The most basic example and therefore a good place to start getting acquainted with the library is the Triangle example. It adds simplices of a triangle one by one, and then (in case of a Triangle zigzag example), removes them one by one.
The simplest example that instead of specifying the complex explicitly, constructs it from the input point set is the Alpha shape example. The example reads points from a file, determines their dimension dynamically (based on the number of coordinates in the first line of the file), and then constructs an alpha shape and outputs its persistence diagram.
Another example that follows a similar strategy is the computation of the Vietoris-Rips complex. Since only pairwise distances are required it works with points in arbitrary dimension. (Of course, in dimensions 2 and 3 the complexes are much larger than those for the Alpha shape example).
One may use persistent cohomology algorithm to extract persistent cocycles, turn them into harmonic cocycles, and use them to parametrize the input point set; for details see [dSVJ09]. The explanation of how to use Dionysus to achieve this is available.
A simple example of computing persistence of a lower-star filtration is in
examples/pl-functions/lsfiltration.py.
A C++-only, but useful example is computation of a vineyard of piecewise straight-line homotopy of piecewise-linear functions.